Mary and Nancy can each perform a certain task in m and n

Question

Mary and Nancy can each perform a certain task in m and n hours, respectively. Is m<n?

(1) Twice the time it would take both Mary and Nancy to perform the task together, each working at their respective constant rates, is greater than m.

(2) Twice the time it would take both Mary and Nancy to perform the task together, each working at their respective constant rates, is less than n.

KarishmaB · Accepted Answer

You can also solve this logically. Use numbers if it is tougher with variables.

Say, together they take 30 mins to complete the work. If both were working at the same rate, each would have taken 60 mins (so that working together would reduce the time taken by half). If one works faster and the other slower, the faster one will take less than 60 mins and the slower one will take more than 60 mins.

Statement 1 says that m (time taken by Mary) is less than 60 mins. This means rate of work of Mary is more than that of Nancy. So m < n. Sufficient.

Statement 2 says that n (time taken by Nancy) is more than 60 mins. This means rate of work of Nancy is less than that of Mary. So n > m. Sufficient.

Answer D

GyanOne · Answer

Using statement (1),
2mn/(m+n) > m
As m and n are both positive,
=> 2mn > m^2 + mn
=> mn > m^2
=> n > m
Sufficient.

Using statement (2),
2mn / (m+n) < n
=> 2mn < mn + n^2
=> mn < n^2
=> m < n
Sufficient.

(D) is the answer.

LifeChanger · Answer

D..
A results in n>m
B results in m<n

Saurajm · Answer

Can you let me know how do you get the expression 2mn/(m+n)??
Thanks

https://gmatclub.com/static/posted_from.png     Posted from    GMAT ToolKit

shinna · Answer

D is the correct answer.

1- Leads for n>m
2- Leads for m<n

Using either of them the question can be answered

KarishmaB · Answer

Mary takes m hrs to complete a job and Nancy takes n hrs. Together, how long will they take?

Time taken by Mary = m hrs
Time taken by Nancy = n hrs

Time taken is inverse of Rate of Work which we get from the expression Work = Rate * Time. If work done is 1, Rate = 1/Time. Hence,

Rate of work of Mary = 1/m
Rate of work of Nancy = 1/n
When they are working together, their rate of work = 1/m + 1/n = (m+n)/mn
Time taken by them together = mn/(m+n)

2mn/(m+n) is 'Twice the time it would take both Mary and Nancy to perform the task together, each working at their respective constant rates'

GMATmission · Answer

D it is.

The basic equation for RTD problems is work = rate X time
Let work be 1, so the rate by which Mary works per hour is 1/m
Similarly, the rate by which Nancy works per hour is 1/n

Statement 1: 
With the information given in the question and using the RTD formula, the combined time to complete the same work (say 1) can be calculated as:

1= (1/m+1/n)t (t is the time)
Therefore, t = mn/(m+n)

Now, according to the statement, 2mn/(m+n) > m
 Here we need to understand that m can not be negative. So we can divide both sides of the equation by m . By solving the equation we get, 
n>m
Hence Sufficient.

Statement 2:
We can prove the statement to be true as Statement 1.
Hence Sufficient

Both statements are sufficient, so the answer is D.

Magdi · Answer

Since the question asked " Is m<n?", I see we can get both answers YES and NO for the question . Because we do not have enough information, using all given info in 1 and 2 may result in both m>n and n>m.... think about it
confusing ? 
All posted solutions assume that Mary's rate is 1/m (1 over m) and Nancy's rate is 1/n, but what is Mary's rate would be m or 2/m or ... so we may have different answers in these diff. cases, do not we ?

KarishmaB · Answer

The question tells you that Mary takes m hours to complete 1 work.

We know

Work = Rate * Time

1 = Rate * m

Rate = 1/m

So the question tells you that Mary's rate is 1/m only.

manishtank1988 · Answer

My 2 cents: my 2 cents.png

gulyaa · Answer

We could probably use smart numbers here and get D as well. If we just use 1 and 2 hours -> rates are 1/2 and 2/2 -> combined rate is 3/2 -> combined time 2/3 hours. 1. twice the time is greater than m, so 2/3*2 = 4/3 or 1.33, so 1< 1.33 < 2. We can conclude that 1 = m and 2 = n, so n>m 2. twice the time is less than n, so the same, 2/3*2 = 4/3 or 1.33, so 1< 1.33 < 2 We can conclude that 1 = m and 2 = n, so n>m

GMAT0010 · Answer

Twice the time means "half the rate" We can't add times directly, so we convert them into rates and then add them. So we get (1/m+1/n) < 2/m and (1/m+1/n) > 2/n Answer is D Hope this helps!

ScottTargetTestPrep · Answer

Solution: Mary’s rate is 1/m, and Nancy’s rate is 1/n. Their combined rate is 1/m + 1/n = (n + m)/mn. Since time is the inverse of rate, their combined time to perform the task is mn/(n + m). We need to determine if m < n. Statement One Alone: Twice the time it would take both Mary and Nancy to perform the task together, each working at their respective constant rates, is greater than m. We are told that twice the combined time is greater than Mary’s time alone. We express this as: 2mn/(n + m) > m 2mn > nm + m^2 mn > m^2 n > m The answer to the question is yes. Statement one is sufficient. Statement Two Alone: Twice the time it would take both Mary and Nancy to perform the task together, each working at their respective constant rates, is less than n. We are told that twice the combined time is less than Nancy’s time alone. We express this as: 2mn/(n + m) < n 2mn < n^2 + mn mn < n^2 m < n The answer to the question is yes. Statement two is sufficient. Answer: D

adityaganjoo · Answer

VeritasKarishma  In considering time = 1/((1/m)+(1/n)) , aren't we assuming that both Mary and Nancy did equal proportion of work?

KarishmaB · Answer

Note how we come to this expression of time taken.

Individual rates of Mary and Nancy are 1/m and 1/n. This means that Mary does (1/m)th of the work every hour and Nancy does (1/n)th of the work every hour. 
When they work together, Mary does (1/m)th of the work and Nancy does (1/n)th in one hour i.e. they together complete (1/m)+(1/n) of the work every hour.

So time taken together to complete 1 full work = 1/(1/m + 1/n) = mn/(m+n)

adityaganjoo · Answer

VeritasKarishma  While I was reverting, I realised where I was going wrong! Thanks!

Oppenheimer1945 · Answer

Mary's rate=a Nancy's rate=b W/a=m; W/b=n; a/W=1/m, b/W=1/n S1)2W/(a+b)>m =>2/(1/m+1/n)>m =>2mn/(m+n)>m =>m(2n/(m+n)-1)>0 =>(n-m)/(m+n)>0 =>n>m S2) 2W/(a+b) 2mn/(m+n) 2m/(m+n)-1<0 => (m-n)/(m+n)<0 =>m

SamCodes · Answer

Basically this problem is asking if Mary is faster than Nadir.
Now, let us consider below scenarios:
Scenario 1: What if both Mary and Nadir worked at the same speed, that is, m=n. Then the twice time (t) taken by both working together to complete the task would be equal to t=m=n.
Scenario 2: If m>n, then twice time (t) taken by both of them working together is n<t<m
Scenario 3: If m<n, then twice time (t) taken by both of them working together is m<t<n

Hope this helps!

Aboyhasnoname · Answer

Thanks Karishma! Your answers help a lot.

I tried to decode the the statement with help of your answer without taking in values to sort of understand what the Statements are saying.....

Statement 1. 
Twice the time together is greater than time m .....Now...This means When M and N are equal the together should be exactly half of m alone....Means Twice of together = m ......Now ..Twice of together is greater than m..means the other person working together is making the together time more...means N's rate is less...if it would have been equal then twice of together should be equal to m....so m< n

Statement 2. 
Similar statement ...Twice of together is less than n...if n were equal to m then twice of together would have been equal to time by n...but no..twice of together is less than n...means m is faster making it possible to complete faster...So Mary is faster than nancy means m < n ....

Mary and Nancy can each perform a certain task in m and n

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Mary and Nancy can each perform a certain task in m and n

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